It can be difficult to balance the coupling shaft at both lower and higher speeds of rotation due to flexibility of the shaft. This will now be explained with reference to FIGS. 1 and 2.
In FIG. 1, a disc 1 is mounted on a shaft 3, and balancing flanges 5, 7 are mounted on the ends of the shaft. The shaft is mounted for rotation about an axis A. The distance between disc 1 and flanges 5, 7 is a and b respectively. The centre of gravity cgdisc of disc 1 is displaced vertically upwards from the axis A1 of disc 1 by a distance edisc (the eccentricity of disc 1). The centre of gravity cgbal of balancing flange 5 is displaced vertically downwards from the axis A2 of flange 5 by a distance ebal (the eccentricity of flange 5). The displacement of the centre of gravity of flange 5 is due to the addition to the flange of balancing weight 9.
For balance of the shaft 3, the centrifugal force CFbal1 acting on flange 5 (as a result of its eccentricity) times the distance a+b from flange 5 to flange 7 must equal the centrifugal force CFdisc acting on disc 1 (as a result of its eccentricity) times the distance b from disc 1 to flange 7, i.e.CFbal1.(a+b)=CFdisc.b  (Equation 1),orMbal.ebal.w2.(a+b)=Mdisc.edisc.w2.b  (Equation 2),where Mbal is the mass of flange 5, Mdisc is the mass of disc 1, and w2 is the angular velocity of rotation squared.
This gives:Mbal.ebal =(b/(a +b)).Mdisc.edisc   (Equation 3),i.e. for balance the mass of balancing flange 5 times the eccentricity of flange 5 must equal b/(a +b) times the mass of disc 1 times the eccentricity of disc 1.
Equation 3 maintains balance at lower speeds of rotation, but at higher speeds an increase in CFdisc (due to the increase in speed) causes flexing of the shaft 3 as shown in FIG. 2 giving rise to a displacement d of the disc 1 from the axis A of rotation (the axis of the shaft 3 is referenced A3 in FIG. 2). The eccentricity of disc 1 is therefore effectively increased by d creating an additional centrifugal force acting on the disc which equals Mdisc.d.w2, resulting in Equation 3 no longer maintaining balance (note, there is negligible displacement of balancing flanges 5, 7 at higher speeds so the only increase in the centrifugal forces acting on flanges 5, 7 at higher speeds is due to the speed increase). Thus, at higher speeds a balancing error occurs which is proportional to the displacement d of the disc 1. It is not possible to balance the shaft 3 fully at both lower and higher speeds of operation due to flexing of the shaft.